## Introduction to Quaternion Calculator:

Quaternion Calculator is an online tool that is used to find the complex numbers that are present in two dimensions to convert into three-dimensional space and converts quaternion to euler online for free.

Quaternion rotation calculator helps to **determine the quaternion** complex number sum, difference, multiplication, conjugate, magnitude, or even matrix and gives solution in seconds.

## What is Quaternion?

Quaternion is a type of group in which **four elements** are present in R4 in three dimensions. Quaternion is a vector set of rotational elements present in 3-D space.

You can calculate arithmetic operations (except division), conjugate, and module problems. It is widely used in mechanics, aerospace, computer graphics, and vision processing.

## Formula Behind Quaternion Multiplication Calculator:

Quaternion basic rules for addition, subtraction, or multiplication for the cartesian problem, where i,j,k are the direction along with x,y,z in three dimensions. The Quaternion Calculator uses the following **formula** to solve the quaternion problems,

$$ q \;=\; w + xi + yj + zk \;where\; i^2 \;=\; j^2 \; =\; k^2 \;=\; ijk \;=\; -1 $$

$$ i^2 \;=\; j^2 \;=\; k^2 \;=\; -1 $$

$$ ij \;=\; -ji \;=\; k, jk \;=\; -kj \;=\; i, -ik \;=\; j $$

## Working Process of Quaternion Rotation Calculator:

Quaternion multiplication calculator uses the easiest method to solve quaternion problems, conjugate, inverse, modulus, etc. You do not need to do any external effort, just add the value and the rest of the calculation will be done automatically.

When you enter your quaternion problem in our quaternion solver. First, it **analyzes the quaternion** question then from the chosen method, it starts calculations.

For addition or multiplication, subtraction, and quotient problems euler to quaternion online solver uses the usual method for solving arithmetic operations. There is only a difference between i, j, k values solved with i, j, k values.

Similarly, when you inverse, conjugate, inverse, or modulus of quaternion q, quaternion to euler calculator substitutes given values in the formula. For the modulus of q, it **adds the a, b, c, and d values** in the modulus formula. For conjugate simply change the imaginary component to keep the real component value unchanged.

## Solved Example of Quaternion:

The quaternion calculator can be used to solve quaternion questions but it’s crucial to understand the manual process of calculation. So an **example of a quaternion problem** with a solution is given below which helps you to understand the calculation process,

### Example:

Solve the following,

$$ r \;=\; a + bi + cj + dk \;and\; r’ \;=\; a’ + b’i + c’j + d’k $$

**Solution:**

$$ r + r’ \;=\; (a + a’) + (b + b’)i + (c + c’)j + (d + d’)k $$

So the multiplication is,

$$ rr’ \;=\; (a + bi + cj + dk)(a’ + b’i + c’j + d’k) $$

$$ \;=\; aa’ + bb’i^2 + cc’j^2 + dd’k^2 + ab’i + ba’i + cd’jk + dc’kj + ac’j + ca’j + bd’ik + db’ki + ad’k + da’k + bc’ij + cb’ji $$

$$ \;=\; (aa’ - bb’ - cc’ - dd’) + (ab’ + ba’ + cd’ - dc’)i + (ac’ + ca’ - bd’ +db’)j + (ad’ + da’ + bc’ - cb’)k $$

### Example:

Solve the given vectors,

$$ p \;=\; 3 + i - 2j + k, \; \; \; q \;=\; 2 - i + 2j + 3k $$

**Solution:**

$$ p . q \;=\; -2 $$

$$ p \times q \;=\; \biggr| \begin{matrix} i & j & k \\ 1 & -2 & 1 \\ -1 & 2 & 3 \\ \end{matrix} \biggr| $$

$$ -8j - 4j $$

### Example:

Find the Quaternion Subtraction.

$$ q^1 \;=\; 2 + 3i + 4j + 5k , q^2 \;=\; 1 + 2i + 1j + 3k $$

**Solution:**

To subtract one quaternion from another,

$$ q^1−q^2 \;=\; (2−1) + (3i−2i) + (4j−1j) + (5k−3k) $$

$$ \;=\; 1 + i + 3j + 2k $$

$$ q^1 − q^2 \;=\; 1+i+3j+2k $$

So, the difference between the two quaternions is 1 + i + 3j + 2k.

### Example:

To find the inverse, modulus, and conjugacy of a quaternion, let's consider a quaternion.

**Solution:**

q = a + bi + cj + dk, where a, b, c, d are real numbers, and i, j, and k are imaginary units.

- 1. Inverse of a Quaternion:

The inverse of a quaternion q is q − 1 and given by:

$$ q − 1 \;=\; ∣q∣^2 $$

- 2. Conjugate of Quaternion:

∣q∣ is the modulus and conjugate (q*) of the quaternion. Conjugate (-q) is the conjugate of the quaternion.

- 3. Modulus of a Quaternion:

The modulus (also known as the norm or absolute value) of a quaternion.

q is denoted as ∣q∣ and is calculated as:

$$ ∣q∣ \;=\; \sqrt{a^2 + b^2 + c^2 + d^2} $$

## How to Use Quaternion Calculator?

Quaternion multiplication calculator has a simple layout that allows you to use it to calculate the quaternion rotation problems in some seconds.

Before entering the input function into the quaternion solver, you must follow some simple steps so that you get a experience during evaluation. These steps are:

- Choose the method type from the given list (sum, difference, multiplication, matrix representation, multiplication, quotient, conjugate, magnitude, vector rotation, quaternion rotation inverse, etc).
**Enter the value of the first quaternion element**Q1 (a, b, c, d) in the input box.- Enter the value of the second quaternion Q2 elements (a, b, c, d) in the input box.
- Enter the second value of y in the input field.
- Review your quaternion function before hitting the calculate button to start the evaluation process.
- Click the “Calculate” button to get the result of your given Quaternion problem.
- If you want to try our quaternion rotation calculator for the first time then you can use the load example option.
- Click on the “Recalculate” button to get refresh page for getting more solutions of Quaternion problems.

## Results of Quaternion Solver:

Euler to quaternion online solver gives you the **solution of given Quaternion method** problem when you give it an input. It provides you with solutions in detail. It contain as:

- Result Option:

Result option gives you a solution for the Quaternion problems of arithmetic operations, inverse, conjugate, etc.

- Possible Steps:

The quaternion calculator give you step by step solution when you click on this option.

## Benefits of Using Euler to Quaternion Online Solver:

Quaternion to euler calculator provides you with multiple benefits whenever you use it to calculate the Quaternion problems and gives a solution in less than a minute. These benefits are:

- Quaternion multiplication calculator is a free-of-cost tool so you can use it to find the Quaternion problem of arithmetic operations and other vector and scalar problems in real time.
- It is a versatile tool that allows you to get the solution of various types of Quaternion problems
- You can check out our quaternion rotation calculator to
**practice with more examples**so that you get a strong hold on the Quaternion concept - It saves you time and effort from doing vector rational calculations.
- It is a trustworthy tool that provides you with every time whenever you use it to calculate Quaternion examples.
- Quaternion calculator provides the solution with a complete process in a step-by-step method so that you get a better understanding of the vector rational set problems